Spin ½, which follows from Dirac’s linearization of the Klein-Gordon equation, is associated with a fermion which in turn is linked to an antisymmetric spin wavefunction under the permutation of particles if other quantum numbers are the same, e.g. energy etc. We argue that the Klein-Gordon equation with a rest mass mo implies a rest frame in which one may observe rotations If one wishes to linearize E, one may write: E = +/- sqrt( p dot p +momo) (c=1). Here p dot p is rotationally invariant, but E and px, py and pz are treated in a different manner even though E and (px,py,pz) appear equally as vector entries of a 4-vector. As a result, one may attempt to linearize the Klein-Gordon equation -EE + pp = -momo to have E and px,py,pz appear on the same footing. This, however, cannot be done using numbers, but requires 4x4 matrices containing 2x2 Pauli matrices as shown by Dirac. We consider the simpler case here of the Pauli 2x2 matrices such that: A = sx p(x) + sy p(y) + sz p(z) and AA = -EE+p dot p. The Pauli matrices are 2x2 and ensure that there is orthogonality among p(x), p(y), p(z) so no cross terms appear in AA. The 2x2 Pauli matrices, however, are linked with more than orthogonality of p(x), p(y), p(z). Given that sxsx=sysy=szsz=1, they carry the same feature of the sqrt function, namely that -1 or 1 squared yields 1. In this case, if one considers the two eigenvectors of sz, namely (1,0) and (0,1), the second has an eigenvalue of -1. One must operate twice with sz i.e. sz sz in order to restore 1. Thus sz is a kind of square root rotation. So far, there has been no mention of antisymmetry which requires at least two particles. If one considers W(spin)= |+>|-> + a |->|+>, where |+>=(1,0) and |->=(0,1), (where a could be +1 or -1), one may ask: Is there a condition on an operator which permutes the first particle and second? Here |n>|m> applies to the first particle in state n and the second in m. We consider the operator sz(1) sz(2), where 1 and 2 denote particles 1 and 2, acting on W(spin). For a single particle, sz acting on |-> returns a -1 eigenvalue, and so for a mixed |+.>|->, an eigenvalue of -1 is also returned. We argue that this represents a physical situation and so the permutation operator should also return the same physical state because Permutation operator * Permutation operator = 1. Thus, we suggest that a permutation should have the same effect as sz(1)sz(2). This suggests that a=-1 i.e. one has |+>|->-|->|+> in order that the permutation operator have the same effect as sz(1)sz(2).
Spin ½, which follows from Dirac’s linearization of the Klein-Gordon equation, is associated with a fermion which in turn is linked to an antisymmetric spin wavefunction under the permutation of particles if other quantum numbers are the same, e.g. energy etc. We argue that the Klein-Gordon equation with a rest mass mo implies a rest frame in which one may observe rotations If one wishes to linearize E, one may write: E = +/- sqrt( p dot p +momo) (c=1). Here p dot p is rotationally invariant, but E and px, py and pz are treated in a different manner even though E and (px,py,pz) appear equally as vector entries of a 4-vector. As a result, one may attempt to linearize the Klein-Gordon equation -EE + pp = -momo to have E and px,py,pz appear on the same footing. This, however, cannot be done using numbers, but requires 4x4 matrices containing 2x2 Pauli matrices as shown by Dirac. We consider the simpler case here of the Pauli 2x2 matrices such that: A = sx p(x) + sy p(y) + sz p(z) and AA = -EE+p dot p. The Pauli matrices are 2x2 and ensure that there is orthogonality among p(x), p(y), p(z) so no cross terms appear in AA. The 2x2 Pauli matrices, however, are linked with more than orthogonality of p(x), p(y), p(z). Given that sxsx=sysy=szsz=1, they carry the same feature of the sqrt function, namely that -1 or 1 squared yields 1. In this case, if one considers the two eigenvectors of sz, namely (1,0) and (0,1), the second has an eigenvalue of -1. One must operate twice with sz i.e. sz sz in order to restore 1. Thus sz is a kind of square root rotation. So far, there has been no mention of antisymmetry which requires at least two particles. If one considers W(spin)= |+>|-> + a |->|+>, where |+>=(1,0) and |->=(0,1), (where a could be +1 or -1), one may ask: Is there a condition on an operator which permutes the first particle and second? Here |n>|m> applies to the first particle in state n and the second in m. We consider the operator sz(1) sz(2), where 1 and 2 denote particles 1 and 2, acting on W(spin). For a single particle, sz acting on |-> returns a -1 eigenvalue, and so for a mixed |+.>|->, an eigenvalue of -1 is also returned. We argue that this represents a physical situation and so the permutation operator should also return the same physical state because Permutation operator * Permutation operator = 1. Thus, we suggest that a permutation should have the same effect as sz(1)sz(2). This suggests that a=-1 i.e. one has |+>|->-|->|+> in order that the permutation operator have the same effect as sz(1)sz(2).
After the discovery of id/dt partial and -id/dx partial as energy and momentum operators in quantum mechanics, Dirac linearized the Klein-Gordon equation to obtain 4x4 matrices based on Pauli 2x2 matrices. As a result, a quantum spin degree of freedom emerged from theory, and matched experimental results. Given the Lorentz invariant form: = -EE + p dot p = -momo (c=1), one may mathematically write: E = +/- sqrt( p dot p + momo), which is standard procedure. One may ask: What is wrong with this expression? We argue in this note that it is not incorrect, but rather incomplete mathematics for the reason that one has to insert +/- by hand. We suggest that a complete mathematical description of the square root of a dot product p dot p should automatically give rise to two solutions associated with +/-. Thus, Dirac’s linearization is not simply a procedure for finding spin, but, we suggest, is a more complete mathematical approach to describing the square root of p dot p, which holds even if one does not consider physical situations. Interestingly, nature seems to choose this more complete mathematical description rather than the looser one using the sqrt function and the manually inserted +/- signs.
In two previous notes (Parts 1 and 2), we argued thta spin matrices arise when one attempts to separate a quadratic equation (continuity equation with the Poynting vector for a photon, Klein-Gordon equation for a spin .5 particle) into two factors. These appear as generalized numbers (i.e. matrices) which may be multiplied together without considering vector operations such as the cross or dot product. Here we try to examine the problem in an alternative way. We note first that in the case of the Klein-Gordon equation one has a 4-vector containing (p vector, E) or (r vector, t) to cite two examples. These transform under the unitary Lorentz transformation, i.e. via a unitary 4x4 matrix. A subset, however, namely p(x), p(y),p(z) or x,y,z of this 4-vector transforms physically under rotations in 3-space, i.e. via a 3x3 unitary matrix. Furthermore, there is an invariance under this 3x3 matrix transformation. In particular, objects such as the dot product or cross product do not change. Classically, one has no issue with vector operators in an equation. In quantum mechanics, however, a free particle is described by the wavefunction exp(-iEt+ip dot r) and the operator i d/dt partial and -i grad return E and p (vector). exp(-iEt+ip dot r), however, describes a translational invariance in space and time, i.e. there is a periodicity. This invariance is clearly seen if the complex conjugate of the free particle wavefunction is multiplied by the wavefunction to yield 1 everywhere.Thus one starts with a quadratic equation W*W and linearizes it to obtain the function W(x,t). The operators which act on W are the generators related to the invariance in W*W. For the quantum free particle, these are id/dt and -i grad.. We suggest that if such a scheme exists for this translational invariance in t and r, one should treat the rotational invariance in the Klein-Gordon equation in a similar manner. In other words, there should exist operators and a function associated with rotational invariance. In this case, the operator is a matrix (Pauli matrices) and the function a spinor.
We have pointed out in previous notes that the free particle wavefunction exp(-iEt+ipx) holds for both a photon and particle with rest mass, while the latter has spin ½ and the former, spin 1. We thus suggested that spin is obtained in a very different manner from exp(-iEt+ipx). Here, we start with the Lorentz invariant -EE+ p dot p cc = -momocc ((1)) and argue, as in (1), that it must be linearized to obtain a linear equation in the intrinsic force, namely mo which is linked to inertia, because force acts in a linear form in nature (e.g. F=dp/dt). At this point, however, p is a vector and there is no concept of -ihbar grad. One may mathematically linearize ((1)) using the approach of Dirac (except that now p is a vector of numbers not an operator). This leads to 4x4 matrices with off diagonal Pauli matrices (multiplied by 1 and -1). One may see that the rotational invariant p dot p may be linearized to s dot p (s = three 2x2 Pauli matrices) because these matrices anticommute. Thus, a 2x2 Pauli matrix is associated with rotational invariance. At this point, this Pauli matrix is simply a math entity. In terms of measurement, given that one has matrices, there are also commutation relations which are important. One may consider another math operator linked with rotational invariance, namely d/dphi for a rotation about a z-axis. In classical physics, one does not have such a derivative appearing as one deals with functions. If d/dphi is an operator generating rotations, then d/dx should generate translations, but this is simply math. The “catch” seems to be that if one introduces a probability f(x,p) (which is unusual because P(x)dx = dx/L, where L is an arbitrary length), then f(p1,x)f(p2,x)=f(p3,x)f(p4,x) if p1+p2=p3+p4 (we use 1-dimension for simplicity). One may, however, shift this equation to x+dx which implies that df(p1)/dx f(p2) + f(p1)df(p2)/dx = df(p3)/dx f(p4) +f(p3) df(p4)/dx. A solution which keeps p present is: exp(i C p x). We use i because we want to have equal modulus at all x points. If C=1, the one has exp(i px) with -id/dx as the operator which retrieves p. One may then apply this idea to L = r x p and find that this new operator commutes like the Pauli matrices and gives rise to d/dphi, the rotational operator postulated above. Thus, it seems that one could start with the notion of spin and obtain the notion of exp(-iEt+ipx), we argue and spin being linked to angular momentum, we argue.
In Part 0 (Parts I and II deal with spin) of this note, we argued that one may begin with two flow equations: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) and find that A(x,t) is the classical action for a free particle i.e A(x,t)= L(v) t where L is the classical Lagranian. In particular, after taking the partial derivatives in ((1a)) and ((1b)) one may set x/t=v. Alternatively, one may link ((1a)) and ((1b)) to eigenvalue equations i.e. id/dt exp(i A) = E exp(iA) ((2a)) and -id/dx exp(iA) = p exp(iA) ((2b)). Thus operators appear, in particular one has the momentum operator -id/dx associated with px etc.. These two equations may be incorporated in EE=pp+mm or E= pp/2m (although E and p have different meanings) to obtain the Klein-Gordon and Schrodinger equations. At this point there is no spin. We argued in Part I that spin may be brought in by considering p in ((1a)) as flowing in 3 directions px, py and pz. In such a case one would normally have pi ei where ei is a basis vector, but with the operator equations being used ((2a)) ((2b)), ei is replaced by a basis operator ai (a matrix) such that when pi ai is squared, ai disappears and one obtains pipi = -d/dx d/dx - d/dy d/dy-d/dz d/dz As P. Dirac has shown this leads to 4x4 matrices containing the 2x2 Pauli matrices. As a result, the independent px,py,pz flow is only linked to spin 1/2 . In part II we discussed other spins. Other spin matrices follow different constraints. In this note, we wish to link the idea of basis operators ai with the idea of angular momentum i.e. rxp. In (1), the spin of a fermion is calculated using current from the Dirac equation J (vector) together with r x J i.e. the classical form of angular momentum. In this note, we wish to show the link between this approach and the basis operator one, which does not on the surface, bring in any idea of angular momentum, are the same.
In a previous note (1), we argued that even though photons and particles with rest mass both have the same quantum free particle probability (wavefunction) exp(-iEt+ipx), they have different spins, spin 1 (photon) and spin ½ (fermion like nucleons, electron). We argued that one may find equations for spin matrices by identifying an inherent force associated with the object. For a particle with rest mass, we argued that one chooses rest mass, mo, which is proportional to the force, inertia. For a photon, one chooses the electric field. The key is then to find an equation which is linear in the inherent force and contains p, E (for special relativistic effects and geometry). For a particle with rest mass, the Lorentz invariant -EE + cc p dot p = -momo cccc is linearized as Dirac showed. For a photon, one seeks an equation linear in the electric field. One may linearize d/dt partial (.5eo El dot El + .5/uo B dot B) + 1/uo grad dot (El x B) = 0, where El=electric field and B=magnetic field. For a particle with rest mass, one must retain the identity of the rest mass when forces interact. If one has two identical mo’s with the same center-of-mass, one must somehow distinguish between the two to deliver an impulse hit to only one. This may be done by using a magnetic field and having one particle have spin up, and the other, spin down. This leads to the Pauli exclusion principle. Using the Pauli exclusion principle in a two body elastic scattering problem, one may easily establish time reversal balance by: p(e1)p(e2) {1-p(e3)}{1-p(e4)} = p(e3)p(e4) {1-p(e1)}{1-p(e2)} ((1)). From this one may obtain the Fermi-Dirac distribution. The question becomes: What does one do for photons? We argue that photons do not interact in the same manner as two particles with rest mass which elastically scatter. Instead we consider photons interacting with electrons in atoms. We note that in the case of particles with rest mass, there is a grouping of two energies, i.e. e1,e2 on the LHS side of ((1)), and e3,e4, on the RHS. We keep this notion of grouping, but note that e1, and e2 are not being converted into e3, e4 photons. Rather, all four types of photons are present and may interact with the electrons in atoms. Thus, we consider the sum of the probabilities for the e1,e2 grouping, namely (e1,e2) + (e1,e2) e3 + (e1,e2) e4 + (e1,e2) e3, e4. We argue that the sum of probabilities for this set of groupings should equal the sum of probabilities for the grouping of (e3,e4), i.e. (e3,e4)+ (e3,e4) e1 + (e3,e4) e2 + (e3,e4) e1,e2. This is equivalent to: p(e1)p(e2) {1+p(e3)}{1+p(e4)} = p(e3)p(e4) {1+p(e1)}{1+p(e2)}. From this, the Bose-Einstein distribution follows.
Physics is based on experimental observables. Some of these observables are functions of other observables. For example, mass is an observable associated with inertia or gravity and velocity is classically measured, but the product mov yields nonrelativistic momentum which may be measured independently through an impulse hit with a target. A further issue with measurement is the case of vectors. For a vector pointing in one direction, one may orient the z axis to lie along this direction. If one always does this, momentum appears as a scalar, but one has to continuously rotate the co-ordinate system to account for different orientations. In this note, we are interested in spin which is linked to a kind of rotation and so we focus on the momentum vector, but only consider a function of momentum which yields a linear E, free particle energy. Thus, for the Lorentz invariant: -EE + p dot p = -momo, we consider the nonrelativistic limit: E = p dot p /2mo. In quantum mechanics, a free particle is represented by the complex probability (wavefunction) exp(ip dot r).This function has physical implications as it manifests itself in n-slit interference experiments so it is physical (not a mathematical convenience), but at the same time is responsible for conservation of momentum in x,y,z because Integral dx exp(ip1x) exp(ip2x) = 0 if p1 is not equal to p2. A physical function (property) through orthogonality creates an important relationship, namely conservation of momentum. We suggest that a similar idea may be at work in a relationship linking a linear E (free particle energy) to a function of momentum. We start with a linear E related to a function of p and only focus on the p vector because if it is treated as a number, the co-ordinate axis must rotate to account for its orientation. We only consider a situation in which p(x), p(y) and p(z) behave in an independent manner in the function of p (i.e. in the same manner as p itself) and suggest that instead of using vector analysis, there should exist a physical co-ordinate axis for each projection p(x), p(y) and p(z) if they are independent. We argue that p dot p = p(x)p(x) + p(y)p(y)+p(z)p(z) does not contain cross terms such as p(x)p(y). This is a kind of orthogonality which we associate with a rotational kind of operator linked with the p(i) operator -id/dx(i). In other words, we seek: p vector = generalized number = p(x)sx+p(y)sy+p(z)sz, where s(i) are generalized numbers or matrices, so that when this multiplied by itself, it yields p dot p. It turns out that the s(i) are the 2x2 Pauli matrices. In the case of a photon, there is no nonrelativistic limit and a linear E equation related to p is: E= |p|c. This equation is not independent in p(x), p(y), p(z) i.e. one cannot write p(x)sx+p(y)sy+p(z)sz an obtain E = {p(x)sx+p(y)sy+p(z)sz} c. Thus we look for another equation containing a linear E and a function of p. In this case, we use energy and momentum densities as given by the continuity equation: d/dt partial energy density + d grad dot ElxB =0 where d is a constant and El and B electric and magnetic 3-vectors. As in the above analysis, one wishes to have a momentum operator linked with a kind of rotational operator which allows one to eliminate vector math and use generalized numbers. In such a case, grad in the continuity equation becomes the momentum vector and eijk (Levi-Civita symbol in the cross product) becomes the spin matrix. Thus in both cases, spin governs the form of the function containing momentum which yields a linear E. In other words, it is not momentum alone which yields the relationship between a linear E (free particle energy) and momentum.